Generally, one omits parentheses in formulas, when there is no ambiguity. For example, a formula (φ) can be simply written φ. As such, the parentheses are also called the auxiliary symbols.

2.

The other logical symbols are obtained in the following way :

φ∧ψ

:=def⁢¬⁡(¬⁢φ∨¬⁢ψ)

⁢φ⇒ψ

:=def⁢¬⁢φ∨ψ

φ⇔ψ

:=def(φ⇒ψ)∧(ψ⇒φ)

⁢∀x⁢(φ)

:=def⁢¬⁡(∃x⁢(¬⁢φ))

where φ and ψ are formulas. All logical symbols are used when building formulas.

3.

In the literature, it is a common practice to write Σω⁢ω for FO⁡(Σ). The first subscript means that every formula in FO⁡(Σ) contains a finite number of ∨’s (less than ω), while the second subscript signifies that every formula has a finite number of ∃’s. In general, Σα⁢β denotes a language built from Σ such that, in any given formula, the number of occurrences of ∨ is less than α and the number of occurrences of ∃ is less than β. When the number of occurrences of ∨ (or ∃) in a formula is not limited, we use the symbol ∞ in place of α (or β). Clearly, if α and β are not ω, we get a language that is not first-order.

First Order Languages as Formal Languages

If the signature Σ and the set V of variables are countable, then S⁢(Σ),T⁢(Σ), and F⁢(Σ) can be viewed as formal languages over a certain (finite) alphabetΓ. The set Γ should include all of the logical connectives, the equality symbol, and the parentheses, as well as the following symbols

R,F,V,I,#,

where they are used to form words for relation, formula, and variable symbols. More precisely,

•

V⁢In⁢# stands for the variable vn,

•

R⁢In⁢#⁢Im⁢# stands for the m-th relation symbol of arity n, and

•

F⁢In⁢#⁢Im⁢# stands for the m-th function symbol of arity n,

where m,n≥0 are integers. The symbol # is used as a delimiter or separator. Note that the constant symbols are then words of the form F⁢#⁢Im⁢#. It can shown that S⁢(Σ),T⁢(Σ) and F⁢(Σ) are context-free over Γ, and in fact unambiguous.